# Find the limits. lim_{xrightarrow-2} g(f(x)) f(x)=x+7 g(x)=x^2

Question
Limits and continuity
Find the limits. $$\lim_{x\rightarrow-2} g(f(x))$$
$$f(x)=x+7 g(x)=x^2$$

2020-11-15
Given $$f(x) = x+7$$ and $$g(x) = x^2$$ ,
To find limit of g(f(x)) first find the expression for g(f(x)) and then apply limit to the resulting expression.
We can g(f(x)) as follow:
$$g(f(x))=g(x+7)$$
$$g(x+7)=(x+7)^2$$
$$=x^2+2(7)(x)+7^2$$
$$=x^2+14x+49$$
Therefore, $$g(f(x))=x^2+14x+49$$
We can the limit for the expression g(f(x)) as follows:
$$\lim_{x\rightarrow-2}g(f(x))=\lim_{x\rightarrow-2}(x^2+14x+49)$$
$$=(-2)^2+14(-2)+49$$
$$=4-28+49=25$$
Therefore $$\lim_{x\rightarrow-2}g(f(x))=25$$

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